Working With Decimal Numbers These notes are designed to help you understand and use some of the mathematical tools that will arise during your studies. You are welcome to visit the Maths Learning Centre in person whenever you feel the need. Our Drop-In Centre in Hub Central is open 10.00am to 4.00pm Monday to Friday during teaching periods, mid-semester breaks and exams (location and contact details are on the next page). c 2014 Maths Learning Centre The University of Adelaide (based on an original document produced in 2005) Maths Learning Centre Level 3, Hub Central North Terrace Campus Phone: 8313 5862 Email: mathslearning@adelaide.edu.au Web: www.adelaide.edu.au/mathslearning
Working With Decimal Numbers/2 Fractions, Decimals and Percentages A fraction like 4 can be thought of as 4 out of 50. Doubling both figures produces the 50 fraction 8 or 8 out of 100. It s easy to see that they both represent the same thing. 100 The word percent means per hundred or out of 100 so both versions of this fraction represent 8%. In fact, so do all of these fractions too: 4 50, 8 100, 2 25, 40 500, 800 10, 000, etc. What if the fraction is more complicated, such as 5 ( 5 out of 8 )? If you can t see a simple 8 way to create a bottom line of 100, convert the fraction to a decimal first. This can be done on a calculator (or, if you like that sort of thing, long division). 5 8 = 0.625. If you are unsure about what this means, the various digits are described here: tens units... 51.359... thousandths hundredths tenths The decimal 0.625 is read zero point six two five and consists of 6 tenths, 2 hundredths and 5 thousandths. In other words, it s bigger than a half and smaller than one. The decimal form can be thought of as a number out of 1 so, to convert this to a percentage ( out of 100 ), simply multiply it by 100. If you try this on a calculator you will get 62.5% Notice that the digits remain the same but the decimal point has moved two places to the right. This is one of the advantages of our base ten number system (ie. ten digits). The topic Working With Decimals discusses this concept further. Example: Convert 1 3 to a percentage.
Working With Decimal Numbers/3 On a calculator we get 1 3 = 0.333..., which is known as a recurring decimal. Multiplying by 100 we get 1 3 = 0.333... or 33.3... % Example: On Census night 2001, there were 18,972,350 people in Australia (including overseas visitors) compared with 17,892,423 in 1996. By what percentage did the population grow in this time? Taking the ratio of 2001 to 1996 we get 18, 972, 350 17, 892, 423 = 1.06035666... from the calculator. This top heavy fraction represents a number larger than 1 as you can see from the number (1) in the units position of the decimal form. Multiplying by 100 we see that the size of the population in 2001 was 106.035666...% of its size in 1996. This says that the population of Australia in 2001 was a bit over 106% of the population size in 1996. In other words, the population grew by a bit over 6% between 1996 and 2001. Exercise 1 Convert the following fractions to percentages: (a) 19 100 (b) 1 4 (c) 3 5 (d) 8 9 (e) 109 110 (f) 5 4 (g) 111 110 (h) 1 200 389 Adelaide residents aged 15 to 24 died between 1996 and 1999. Of these, 282 were male. (i) What percentage of these deaths were male? 67.1% of these 389 deaths were caused by injury or poisoning. (j) How many actual deaths does this percentage represent?
Working With Decimal Numbers/4 The Power of Tens Numbers that we work with can be become very large. For example, there were 4,647 people in Australia aged 95 and over at the 2001 Census. In words, we might say over four and a half thousand so that the number is easier to describe. This changes the unit of measurement from people to thousands of people. A useful feature of our base 10 number system is that, if we change the units of a number to a power of 10 (eg. hundreds, thousands, tens of thousands, etc), the actual digits remain the same and the decimal point moves to the left a number of steps equal to the power of 10 involved. Example: One thousand can be written as 1, 000 or 10 3 ( a one with three zeros after it ). Hence 4, 647 = 4.647 thousand. The decimal point (assumed to be after the 7) has moved back 3 steps. Example: There were 18,769,249 people in Australia at the 2001 Census, meaning more than 18 and a half million : 18, 769, 249 = 18.769249 million. Since one million is 1, 000, 000 or 10 6, the decimal point has moved back 6 steps. Exercise 2 Express the following numbers in the units given and state the power of 10 involved: (a) 10,312 in thousands (b) 9,502,703 in millions (c) 1,027,015,247 in millions (d) 1,027,015,247 in billions (1 billion = 1,000,000,000)
Working With Decimal Numbers/5 Exponential/Scientific Notation? If we use powers of 10 to move the decimal point in large numbers to sit next to the first digit, the number is said to be in Exponential or Scientific Notation: 4, 647 18, 769, 249 = 4.647 1, 000 = 4.647 103 = 1.8769249 10, 000, 000 = 1.8769249 107 In this form, the power of 10 gives an indication of the size or magnitude of the number. The same can be done for very small numbers such as 0.0018, by moving the decimal point the other way and using negative powers of ten: 1 0.001 8 = 1.8 thousandths = 1.8 1, 000 0.000003 45 = 3.45 millionths = 3.45 1 1, 000, 000 = 1.8 10 3 = 3.45 10 6 If you are unsure about the decimal positions used above (thousandths, millionths, etc) see Appendix A. In the following exercises it may help to use the table of powers of ten and the various ways of writing them in Appendix A. Exercise 3 Convert the following numbers to Scientific Notation: (a) 38,000 (b) 0.04 (c) 0.000019 (d) 0.010004 (e) 710.5 (f) 1,963.09 Convert the following numbers to ordinary decimals: (g) 2.65 10 3 (h) 1.57 10 4 (i) 1.5 10 8 (j) 5.005 10 2
Working With Decimal Numbers/6 Rates If you drive 180 kilometres in 3 hours, your average speed was 180 km 3 hours = 60 km per hour (or km/h ). Similarly, there were 128,932 deaths registered in Australia in 2001 and 18,769,249 people in Australia on Census Day 2001. From this information we can estimate the crude death rate : 128, 932 deaths 18, 769, 249 people = 0.006869321 deaths per person. It makes sense to speak of speeds in per hour terms but deaths per person doesn t present the information in a very useful form. In this case it is better to change the units to, say, deaths per 100,000 people. This can be done simply by multiplying the deaths per person figure by 100,000 people: 0.00689321 100, 000 = 0.00686 9321 105 = 686.9321 deaths per 100,000 people. Exercise 4 There were 9,502,703 women in Australia on Census Day 2001 and 61,709 female deaths registered in the same year. Work out the crude death rate per woman and convert it to: (a) deaths per 100,000 women (b) deaths per 1,000 women In a hypothetical area, 12 cases of a particular syndrome were reported last year out of a population of 125,000. Work out: (c) the incidence rate per 100,000 people.
Working With Decimal Numbers/7 Scientific/Exponential Notation on a Calculator or Computer If a calculation produces a number too large or too small for your scientific or graphic calculator to display directly, it will switch to scientific/exponential notation. If you want to enter a number in scientific/exponential notation, the EXP key is a good shortcut. For example, to enter 1.2 10 5, type 1 2 EXP 5 Example: To enter 5.34 10 6, type 5 3 4 EXP 6 +/ Note that the +/ key comes after the number. Some calculators have a ( ) key instead which allows you to enter the negative sign in the natural place: 5 3 4 EXP ( ) 6 In computer spreadsheets and output, options for presenting numbers in scientific/exponential notation are limited. If you see something like 4.56789E05, it stands for 4.56789 10 5.
Working With Decimal Numbers/8 Rounding Numbers Calculations often produce answers with many decimal places. For example, 2 people out of every 3 represents an exact percentage of 66.6666...% but, for simplicity we would probably round this off to the first decimal place or the nearest whole number. The process of rounding numbers is as follows: (i) Decide where you want your rounded value to stop. (ii) Look at the next digit on the right. If it is a 5, 6, 7, 8 or 9 then increase the last digit of the rounded value by 1. 0, 1, 2, 3 or 4 then do nothing. To round 66.6666... to one decimal place, we look at the second decimal place. Since this is a 6 we round up the first decimal place: Answer: 66.7%. Examples: (1) Round 7.387056 to two decimal places. Solution: The third decimal place is 7, so round up to 7.39. (2) Round 4639.49 to the nearest whole number. Solution: The first digit after the decimal point is 4, so leave the previous digit alone: 4639. (3) Round 5.96333 to one decimal place. Solution: The second decimal place is 6 but, since we can t replace the 9 by 10 directly, we replace 9 by 0 and increase the next digit on its left by 1. The rounded value is 6.0. (4) Round 9.96333 to one decimal place. Solution: If we have a number ending in a string of 9 s, then we repeat the last step above as many times as necessary. In this case the rounded value is 10.0.
Working With Decimal Numbers/9 Exercise 5 Round: (a) 3.812 to two decimal places (b) 4.56 to one decimal place (c) 105.5 to the nearest whole number (d) 0.0034 to two decimal places (e) 15.0999 to one decimal place (f) 95.4999 to the nearest whole number (g) 95.9999 to three decimal places (h) 0.00444... to three decimal places
Working With Decimal Numbers/10 Appendix A: Powers of 10 and Decimal Places millions 1, 000, 000 10 6 hundred thousands 100, 000 10 5 ten thousands 10, 000 10 4 thousands 1, 000 10 3 hundreds 100 10 2. tens 10 10 1 (anything to the power 1 is itself) ones 1 10 0 (anything to the power 0 is 1) tenths 0.1 10 1 1 10 hundredths 0.01 10 2 1 100 thousandths 0.001 10 3 1 1,000 ten thousandths 0.0001 10 4 1 10,000 hundred thousandths 0.00001 10 5 1 100,000 millionths 0.000001 10 6 1 1,000,000. Example of decimal places: tens units... 51.359... thousandths hundredths tenths
Working With Decimal Numbers/11 Appendix B: Short Answers to Exercises Answers 1 (a) 19% (b) 25% (c) 60% (d) 88.88... % (e) 99.0909... % (f) 125% (g) 100.9090... % (h) 0.5% (ie. half of 1% ) (i) 72.49... % (j) 0.671 389 = 261.019 so there were 261 deaths. (The answer was not a whole number because the quoted 67.1% was rounded off. See Working With Decimal Numbers: Rounding Numbers. Answers 2 (a) 10.312 thousand (1, 000 = 10 3 ) (b) 9.502703 million (1, 000, 000 = 10 6 ) (c) 102.7015247 million (d) 1.027015247 billion (1, 000, 000, 000 = 10 9 ) The number in (b) is the number of women in Australia according to the 2001 Census. The number in (c) and (d) is the provisional population of India according to their 2001 Census. It meant that India officially became the second country in the world after China to exceed one billion people. Answers 3 (a) 3.8 10 4 (b) 4 10 2 (c) 1.9 10 5 (d) 1.0004 10 2 (e) 7.105 10 2 (f) 1.96309 10 3 (g) 2,650 (h) 0.000157 (i) 150,000,000 (j) 0.05005
Working With Decimal Numbers/12 Answers 4 Crude death rate per woman = 0.0064938... (a) 649.38... deaths per 100,000 women (b) 6.4938... deaths per 1,000 women Note: Normally we would round off to say 649.4. See Tip: Rounding Numbers for more details. (c) 9.6 cases per 100,000 people Answers 5 (a) 3.81 (b) 4.6 (c) 106 (d) 0.00 (e) 15.1 (f) 95 (g) 96.000 (h) 0.004
Working With Decimal Numbers/13 Appendix C: Detailed Answers to Exercises Detailed Answers 1 Convert the following fractions to percentages: 19 1 (a) (b) 100 4 = 25 100 = 19 out of 100 or 19% = 25 out of 100 or 25% (or type 1 4 into a calculator to give 0.25) (c) 3 5 = 60 100 = 60% (d) 8 9 (or 3 5 on a calculator gives 0.6) = 0.888... on a calculator = 88.8...% (e) 109 110 = 0.990909... on a calculator = 125 100 (f) 5 4 or 125% = 99.0909... % (or 5 4 on a calculator gives 1.25) (g) 111 110 (h) 1 200 = 1.009090... on a calculator = 0.005 on a calculator = 100.9090...% = 0.5% (ie. half of 1% ) 389 Adelaide residents aged 15 to 24 died between 1996 and 1999. Of these, 282 were male. (i) What percentage of these deaths were male? 282 males 389 deaths = 0.72493... = 72.49... % 67.1% of these 389 deaths were caused by injury or poisoning. (j) How many actual deaths does this percentage represent? 67.1% of 389 = 0.671 389 = 261.019, so there were 261 deaths. (The answer was not a whole number because the quoted 67.1% was rounded off. See Rounding Numbers.)
Working With Decimal Numbers/14 Detailed Answers 2 Express the following numbers in the units given and state the power of 10 involved: (a) 10,312 in thousands (b) 9,502,703 in millions Power of 10 used is 3 so move the decimal point back 3 places: Power of 10 used is 6 so move the decimal point back 6 places: 10.312 thousand 9.502703 million (c) 1,027,015,247 in millions (d) 1,027,015,247 in billions (1 billion = 1,000,000,000) Power of 10 used is 6 so move the decimal point back 6 places: Power of 10 used is 9 so move the decimal point back 9 places: 102.7015247 million 1.027015247 billion
Working With Decimal Numbers/15 Detailed Answers 3 Convert the following numbers to Scientific Notation: (a) 38, 000 (b) 0.04 (c) 0.00001 9 = 3.8 10 4 = 4 10 2 = 1.9 10 5 (d) 0.01 0004 (e) 710.5 (f) 1,963.09 = 1.0004 10 2 = 7.105 10 2 = 1.96309 10 3 Convert the following numbers to ordinary decimals: (g) 2.65 103 (h) 1.57 10 4 = 2,650 = 0.000157 (i) 1.5 108 (j) 5.005 10 2 = 150,000,000 = 0.05005
Working With Decimal Numbers/16 Detailed Answers 4 There were 9,502,703 women in Australia on Census Day 2001 and 61,709 female deaths registered in the same year. Work out the crude death rate per woman and convert it to: 61, 709 9, 502, 703 = 0.0064938... (a) deaths per 100,000 women (b) deaths per 1,000 women 0.0064938... 100, 000 0.0064938... 1, 000 = 0.00649 38... 105 = 0.006 4938... 103 = 649.38... deaths per 100,000 women = 6.4938... deaths per 1,000 women In a hypothetical area, 12 cases of a particular syndrome were reported last year out of a population of 125,000. Work out: (c) the incidence rate per 100,000 people. The incidence rate per person is people is 0.000096... 100, 000 = 0.00009 6... 105. = 9.6. 12 125, 000 = 0.000096..., so the rate per 100,000 Note: It s a good idea to check if your answer looks reasonable. In this case, 12 out of 125,000 is roughly 10 out of 100,000.
Working With Decimal Numbers/17 Detailed Answers 5 Round: (a) 3.812 to two decimal places (b) 4.56 to one decimal place The third decimal place is 2 so The second decimal place is 6 so leave the second decimal place increase the first decimal place by unchanged: 3.81 one: 4.6 (c) 105.5 to the nearest whole number (d) 0.0034 to two decimal places The first decimal place is 5 so increase the units by one: 106 The third decimal place is 3 so leave the second decimal place unchanged: 0.00 (Strange but true! This says zero to two decimal place accuracy.) (e) 15.0999 to one decimal place (f) 95.4999 to the nearest whole number The second decimal place is 9 so The first decimal place is 4 so increase the first decimal place by leave the units unchanged: 95 by one: 15.1 (g) 95.9999 to three decimal places (h) 0.00444... to three decimal places The fourth decimal place is 9 so The fourth decimal place is 4 so raise the third decimal place by one. leave the third decimal place Since this 9, replace it by 0and raise unchanged: 0.004 the second decimal place by one. (Don t be put off by the recurring Repeat as required!: 96.000 digit.)